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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 281 15 "Revised Simplex" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 17 "Define functions:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 79 "E_2 := proc(n, i, alpha)\n   mulrow(array(iden
tity, 1..n,1..n), i, alpha);\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "Pivot:=proc(A,i,j) linalg[pivot](E_2(rowdim(A),i,1/A[
i,j]) &* A,i,j) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&PivotGR6%%
\"AG%\"iG%\"jG6\"F*F*-&%'linalgG6#%&pivotG6%-%#&*G6$-%$E_2G6%-%'rowdim
G6#9$9%*&\"\"\"F=&F:6$F;9&!\"\"F:F;F@F*F*F*" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 37 "Text, p. 121, no. 20:                " }{TEXT 272 4 " m
ax" }{TEXT -1 3 " \{ " }{TEXT 256 1 "c" }{TEXT -1 3 "^T " }{TEXT 257 
1 "x" }{TEXT -1 6 " :  A " }{TEXT 258 1 "x" }{TEXT -1 4 " <= " }{TEXT 
259 1 "b" }{TEXT -1 2 ", " }{TEXT 260 1 "x" }{TEXT -1 3 " >=" }{TEXT 
261 2 " 0" }{TEXT -1 7 " \} with" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 87 "A:=matrix(3,4,[2,1,3,1,2,3,0,4,3,1,2,0]);b:=vector(3,
[8,12,18]);c:=vector(4,[1,2,1,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"AG-%'matrixG6#7%7&\"\"#\"\"\"\"\"$F+7&F*F,\"\"!\"\"%7&F,F+F*F." }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'vectorG6#7%\"\")\"#7\"#=" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG-%'vectorG6#7&\"\"\"\"\"#F)F)" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "cA:=stackmatrix(-c,A);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cAG-%'matrixG6#7&7&!\"\"!\"#F*F*7&
\"\"#\"\"\"\"\"$F.7&F-F/\"\"!\"\"%7&F/F.F-F1" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 89 "Introduce slack variables (the slack columns now appear
 in front of the original columns)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "A1:=augment(array(1..4,1..4,identity),cA);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#A1G-%'matrixG6#7&7*\"\"\"\"\"!F+F+!\"\"!
\"#F,F,7*F+F*F+F+\"\"#F*\"\"$F*7*F+F+F*F+F/F0F+\"\"%7*F+F+F+F*F0F*F/F+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Abc:=augment(vector(4,[
0,0,0,0]),A1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$AbcG-%'matrixG6#7
&7+\"\"!\"\"\"F*F*F*!\"\"!\"#F,F,7+F*F*F+F*F*\"\"#F+\"\"$F+7+F*F*F*F+F
*F/F0F*\"\"%7+F*F*F*F*F+F0F+F/F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
26 "The RHS  b is first column" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 282 
11 "Iteration 0" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 52 "for k from 1 to 3 do Abc[k+1,1]:=b[k] od:evalm(Abc);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7+\"\"!\"\"\"F(F(F(!\"\"!\"#F*
F*7+\"\")F(F)F(F(\"\"#F)\"\"$F)7+\"#7F(F(F)F(F.F/F(\"\"%7+\"#=F(F(F(F)
F/F)F.F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Abc is in memory, and
 is not updated" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "M:=subma
trix(Abc,1..4,2..5);Minv:=inverse(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"MG-%'matrixG6#7&7&\"\"\"\"\"!F+F+7&F+F*F+F+7&F+F+F*F+7&F+F+F+F*
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%MinvG-%'matrixG6#7&7&\"\"\"\"\"
!F+F+7&F+F*F+F+7&F+F+F*F+7&F+F+F+F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 33 "Minv is updated at each iteration" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 288 11 "Iteration 1" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 27 "The reduced costs (top row)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 95 "rc:=vector(9,[]):for k from 1 to 9 do rc[k]:=add(eval
(Minv[1,j]*Abc[j,k]),j=1..4) od:evalm(rc);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'vectorG6#7+\"\"!\"\"\"F'F'F'!\"\"!\"#F)F)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "In the matrix Abc choose  column 6
 to enter" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "q:=6;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"qG\"\"'" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 16 "Update column 1 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 60 "bb:=convert(evalm(Minv &* submatrix(Abc,1..4,1..1)),vector);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bbG-%'vectorG6#7&\"\"!\"\")\"#7\"#=
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Update column q" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "a:=convert(evalm(Minv &* submatrix(
Abc,1..4,q..q)),vector);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG-%'v
ectorG6#7&!\"\"\"\"#F*\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Se
lect row 2 to leave" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p:=2;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"\"#" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 72 "eta:=vector(4,[]):for k from 1 to 4 do eta[k]:
=-a[k]/a[p] od:evalm(eta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vecto
rG6#7&#\"\"\"\"\"#!\"\"F*#!\"$F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "eta[p]:=-eta[p]/a[p]:evalm(eta);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'vectorG6#7&#\"\"\"\"\"#F'!\"\"#!\"$F)" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 13 "The new basis" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 46 "for k from 1 to 4 do M[k,p]:=a[k] od:evalm(M);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&\"\"\"!\"\"\"\"!F*7&F*
\"\"#F*F*7&F*F,F(F*7&F*\"\"$F*F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
63 "The pivot on [6,2]  is obtained by multiplication by the matrix" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "E:=matrix(4,4,[1,0,0,0,0,1
,0,0,0,0,1,0,0,0,0,1]):for k from 1 to 4 do E[k,p]:=eta[k] od:evalm(E)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&\"\"\"#F(\"\"#\"
\"!F+7&F+F)F+F+7&F+!\"\"F(F+7&F+#!\"$F*F+F(" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 67 "The inverse of the basis, Minv,  is updated   by Minv  \+
:=  E * Minv" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Minv:=evalm
(E &* Minv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%MinvG-%'matrixG6#7&
7&\"\"\"#F*\"\"#\"\"!F-7&F-F+F-F-7&F-!\"\"F*F-7&F-#!\"$F,F-F*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 11 "inverse(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'m
atrixG6#7&7&\"\"\"#F(\"\"#\"\"!F+7&F+F)F+F+7&F+!\"\"F(F+7&F+#!\"$F*F+F
(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 283 11 "Iteration 2" }{TEXT -1 1 ":
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Calculate the reduced costs" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "rc:=vector(9,[]):for k fr
om 1 to 9 do rc[k]:=add(eval(Minv[1,j]*Abc[j,k]),j=1..4) od:evalm(rc);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7+\"\"%\"\"\"#F(\"\"#
\"\"!F+F+#!\"$F*F)#!\"\"F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Col
umn 7 enters" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "q:=7;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG\"\"(" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 22 "Update columns 1 and q" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "bb:=convert(evalm(Minv &* submatrix(Abc,1..4,1..1)),v
ector);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bbG-%'vectorG6#7&\"\"%F)
F)\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "a:=convert(evalm
(Minv &* submatrix(Abc,1..4,q..q)),vector);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"aG-%'vectorG6#7&#!\"$\"\"##\"\"\"F+F+#!\"\"F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p:=3;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"pG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
72 "eta:=vector(4,[]):for k from 1 to 4 do eta[k]:=-a[k]/a[p] od:evalm
(eta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7&#\"\"$\"\"%#!
\"\"F)F+#\"\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eta[p]
:=-eta[p]/a[p]:evalm(eta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vecto
rG6#7&#\"\"$\"\"%#!\"\"F)#\"\"\"\"\"##F-F)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 97 "E:=matrix(4,4,[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1]):fo
r k from 1 to 4 do E[k,p]:=eta[k] od:evalm(E);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7&7&\"\"\"\"\"!#\"\"$\"\"%F)7&F)F(#!\"\"F,F
)7&F)F)#F(\"\"#F)7&F)F)#F(F,F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 
"Update inverse" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Minv:=ev
alm(E &* Minv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%MinvG-%'matrixG6
#7&7&\"\"\"#!\"\"\"\"%#\"\"$F-\"\"!7&F0F.F+F07&F0#F,\"\"##F*F4F07&F0#!
\"(F-#F*F-F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 284 11 "Iteration 3" }
{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Calculate the re
duced costs" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "rc:=vector(9
,[]):for k from 1 to 9 do rc[k]:=add(eval(Minv[1,j]*Abc[j,k]),j=1..4) \+
od:evalm(rc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7+\"\"(\"
\"\"#!\"\"\"\"%#\"\"$F+\"\"!F.F.#!\"(F+#F'F+" }}}{EXCHG {PARA 257 "" 
0 "" {TEXT 285 24 "Select column 3 to enter" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 5 "q:=3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG\"\"
$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "bb:=convert(evalm(Minv
 &* submatrix(Abc,1..4,1..1)),vector);" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 59 "a:=convert(evalm(Minv &* submatrix(Abc,1..4,q..q)),vector);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bbG-%'vectorG6#7&\"\"(\"\"$\"\"#F)
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG-%'vectorG6#7&#!\"\"\"\"%#\"
\"$F+#F*\"\"##!\"(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p:=2
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"\"#" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 72 "eta:=vector(4,[]):for k from 1 to 4 do eta[k]
:=-a[k]/a[p] od:evalm(eta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vect
orG6#7&#\"\"\"\"\"$!\"\"#\"\"#F)#\"\"(F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "eta[p]:=-eta[p]/a[p]:evalm(eta);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'vectorG6#7&#\"\"\"\"\"$#\"\"%F)#\"\"#F)#\"\"(F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "E:=matrix(4,4,[1,0,0,0,0,1,0
,0,0,0,1,0,0,0,0,1]):for k from 1 to 4 do E[k,p]:=eta[k] od:evalm(E);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&\"\"\"#F(\"\"$\"\"
!F+7&F+#\"\"%F*F+F+7&F+#\"\"#F*F(F+7&F+#\"\"(F*F+F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 23 "Minv:=evalm(E &* Minv);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%%MinvG-%'matrixG6#7&7&\"\"\"\"\"!#\"\"#\"\"$F+7&F+
F*#!\"\"F.F+7&F+F+#F*F.F+7&F+F+F0F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
286 12 "Iteration 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "rc:
=vector(9,[]):for k from 1 to 9 do rc[k]:=add(eval(Minv[1,j]*Abc[j,k])
,j=1..4) od:evalm(rc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#
7+\"\")\"\"\"\"\"!#\"\"#\"\"$F)#F(F,F)!\"\"#\"\"&F," }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 5 "q:=8;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"qG\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "bb:=convert(e
valm(Minv &* submatrix(Abc,1..4,1..1)),vector);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "a:=convert(evalm(Minv &* submatrix(Abc,1..4,q..q)),ve
ctor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bbG-%'vectorG6#7&\"\")\"
\"%F*\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG-%'vectorG6#7&!\"\"
\"\"$\"\"!\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p:=2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"\"#" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 72 "eta:=vector(4,[]):for k from 1 to 4 do eta[k]:=-a[
k]/a[p] od:evalm(eta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#
7&#\"\"\"\"\"$!\"\"\"\"!#!\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 32 "eta[p]:=-eta[p]/a[p]:evalm(eta);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'vectorG6#7&#\"\"\"\"\"$F'\"\"!#!\"#F)" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 97 "E:=matrix(4,4,[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1
]):for k from 1 to 4 do E[k,p]:=eta[k] od:evalm(E);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'matrixG6#7&7&\"\"\"#F(\"\"$\"\"!F+7&F+F)F+F+7&F+F+
F(F+7&F+#!\"#F*F+F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Minv
:=evalm(E &* Minv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%MinvG-%'matr
ixG6#7&7&\"\"\"#F*\"\"$#\"\"&\"\"*\"\"!7&F0F+#!\"\"F/F07&F0F0F+F07&F0#
!\"#F,F2F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 287 12 "Iteration 5:" }
{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "rc:=vector
(9,[]):for k from 1 to 9 do rc[k]:=add(eval(Minv[1,j]*Abc[j,k]),j=1..4
) od:evalm(rc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7+#\"#G
\"\"$\"\"\"#F*F)#\"\"&\"\"*\"\"!#\"\"(F.F/F/#\"#9F." }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 53 "Optimality condition satisfied. The optimal tab
le is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Abc:=evalm(Minv &
* Abc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$AbcG-%'matrixG6#7&7+#\"#
G\"\"$\"\"\"#F-F,#\"\"&\"\"*\"\"!#\"\"(F1F2F2#\"#9F17+#\"\"%F,F2F.#!\"
\"F1F2#F9F1F2F-F:7+F9F2F2F.F2#\"\"#F,F-F2F87+#\"#MF,F2#!\"#F,F:F-#\"#8
F1F2F2#!#5F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 293 
16 "optimal solution" }{TEXT -1 4 " is " }{TEXT 289 9 "x4 = 34/3" }
{TEXT -1 3 ",  " }{TEXT 290 7 "x6 =  4" }{TEXT -1 2 ", " }{TEXT 291 9 
"x8 = 4/3 " }{TEXT -1 6 " with " }{TEXT 292 20 "optimal value = 28/3" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "74 0 0" 30 }
{VIEWOPTS 1 1 0 1 1 1803 }